Global Behavior of the Difference Equation x n + 1 = ( p + x n - 1 ) / ( q x n + x n - 1 )

Abstract

We study the following difference equation $x_{n+1}=(p+x_{n-1})/(q{x}_n+x_{n-1})$, $n=\mathrm{0,1},\dots ,$ where $p,q\in (0,+\infty )$ and the initial conditions $x_{-1},x_0\in (0,+\infty )$. We show that every positive solution of the above equation either converges to a finite limit or to a two cycle, which confirms that the Conjecture 6.10.4 proposed by Kulenović and Ladas (2002) is true.